Let me point you to the same question that was posted on another forum:

https://answers.acrobatusers.com/Multiple-Users-One-PDF-q22969.aspx

I quote Thom Parker's reply:

Odd, Any number of people aught to be able to open the PDF simultaneously. However, only one person can save the file. This is a feature of the operating system, not Acrobat or the PDF. What is the behavior you are seeing? What exactly is happening?

An alternative way to distribute the files is to create a local website, where everyone can open the form on line. Then setup a server script for recieving the filled/signed forms. That way you avoid the messy issue of reused files on a shared folder.

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